Download series with non-zero central critical value

January 20, 1998
L-SERIES WITH NON-ZERO CENTRAL CRITICAL VALUE
Kevin James
Department of Mathematics
Pennsylvania State University
218 McAllister Building
University Park, Pennsylvania 16802-6401
Phone: 814-865-7527
Fax: 814-865-3735
[email protected]
Abstract. Given a cusp form f of even integral weight and its associated L-function
L(f, s), we expect that a positive proportion of the quadratic twists of L will have nonzero central critical value. In this paper we give examples of weight two newforms whose
associated L-functions have the property that a positive proportion of its quadratic twists
have non-zero central critical value.
1. INTRODUCTION
Suppose that f = n≥1 an (f )q n is a cusp form of weight 2k (k ∈ N). We denote by
L(f, s) the L-function of f . For Re(s) sufficiently large, the value of L(f, s) is given by
P
)
L(f, s) = n≥1 ann(f
and, one can show that L(f, s) has analytic continuation to the
s
entire complex plane. The value of L(f, s) at s = k will be of particular interest to us,
and we will refer to this value as the central critical value of L(f, s).
√
D)/Q, that
Let χD denote the Dirichlet character associated to the extension
Q(
√
∆D
is χD (n) = n , where ∆D denotes
the discriminant of Q( D)/Q. Define the Dth
P
quadratic twist of f to be fχD = n≥1 an (f )χD (n)q n . For any integer D, the L-function
P
a (f )χ (n)
of fχD is the Dth quadratic twist of L(f, s), that is L(fχD , s) = n≥1 n nsD . We
will be interested in determining how often L(fχD , s) has nonzero central critical value as
D varies over all integers. Since χDm2 = χD , we will restrict our attention to the squarefree integers D. We expect that as we let D vary over all of the square-free integers, a
positive proportion of the L-functions L(fχD , s) will have nonzero central critical value.
Indeed, Goldfeld [7] conjectures that for newforms f of weight 2, L(fχD , 1) 6= 0 for 12 of
the square-free integers.
Given an elliptic curve E : y 2 = x3 + Ax2 + Bx + C (A, B, C ∈ Z) with conductor
NE and an integer D, we define the Dth quadratic twist of E to be the curve ED : y 2 =
x3 + ADx2 + BD2 x + CD3 . Let L(ED , s) denote the L-function associated to ED . For
square-free D coprime to 2NE , L(ED , s) is simply the Dth quadratic twist of L(E1 , s).
If f ∈ S2 (N ) is a newform with integer coefficients, we know via the theory of Eichler
and Shimura that there is an elliptic curve E over Q having conductor N so that L(E, s) =
L(f, s). Thus if D is coprime to 2N , then L(ED , s) = L(fχD , s). Also, one knows form
P
1
2
KEVIN JAMES
the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or
that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is
a modular elliptic curve and, if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the
property that a positive proportion of the twists of L(f, s) have nonzero central critical
value, then this implies that a positive density of the quadratic twists ED have rank 0.
There have been many papers which have proved results in this direction. For example,
in [2, 3, 6, 10, 16, 17, 19, 28] one can find general theorems on the vanishing and
nonvanishing of the quadratic twists of a given L-function. These theorems ensure that
an infinite number of the quadratic twists of an L-function associated to a cusp form will
have nonzero central critical value.
In [20], Ono has shown several examples of cusp forms f associated to elliptic curves
such that for a positive density of the primes p, the pth quadratic twist of L(f, s) will have
nonzero central critical value. Ono also proves a theorem which gives sufficient conditions
under which a cusp form associated to an elliptic curve will have this property. Using
methods similar to those of Ono, the author [11] was able to prove that the elliptic curve
Ep : y 2 = x3 − 32p3 has rank 0 for at least 1/3 of the primes p.
Subsequently, Ono and Skinner [22] used the theory of Galois representations to extend
Ono’s theorem to all even weight eigenforms satisfying a very mild hypothesis. In fact
they verify that this hypothesis is satisfied for all modular elliptic curves of conductor
less than or equal to 100.
In a series of two papers [8, 9], Heath-Brown has done an extensive investigation of
the behavior of the 2-Selmer groups associated to the quadratic twists of the congruent
number curve: y 2 = x3 − x. He states as a corollary to one of his theorems that at
least 5/16 of these quadratic twists have rank 0. This result along with the Birch and
Swinnerton-Dyer conjecture implies that at least 5/16 of the quadratic twists of the Lfunction L(E, s) associated to the congruent number curve should have nonzero central
critical value.
Using ideas developed by Frey [5] along with the Davenport-Heilbronn theorem [18],
Wong [27] has shown the existence of an infinite family of non-isomorphic elliptic curves
such that a positive proportion of the quadratic twists of each curve have rank 0. Thus if
we assume the Birch and Swinnerton-Dyer and Shimura-Taniyama conjectures, Wong’s
result would then imply the existence of an infinite family of weight 2 cusp forms {fi }
such that a positive proportion of the twists of each L(fi , s) have nonzero central critical
value.
In section 3 of this paper we exhibit weight 2 newforms F such that L(FχD , 1) 6= 0
for all D in a subset of the square-free natural numbers having positive lower density.
We now describe the first of those results. Let E denote the elliptic curve with equation
y 2 = x3 − x2 + 72x + 368. Then E is a modular curve (it is the −1 twist of X0 (14)). We
let F denote the weight 2 cusp form whose Mellin transform is L(E, s). We then prove
unconditionally:
Theorem 1. L(FχD , 1) 6= 0 for at least 7/64 of the square-free natural numbers D.
In light of Kolyvagin’s work, we have as a corollary to Theorem 1
L-SERIES WITH NON-ZERO CENTRAL CRITICAL VALUE
3
Corollary 2. For at least 7/64 of the square-free natural numbers D, ED has rank 0.
Our proof differs from those of Heath-Brown and Wong in that while they work directly
with the Selmer groups of elliptic curves, our proof uses the theory of modular forms
developed by Shimura and Waldspurger to gain information about the central critical
values of the L-functions associated to elliptic curves. An outline of the proof of Theorem
1 is as follows. Using ideas of Schoeneberg [23] and Siegel [25], we construct a weight
3/2 cusp form f as the difference of the theta functions associated to two inequivalent
ternary quadratic forms Q1 and Q2 which together make up a genus of ternary forms.
This f will be an eigenform for all of the Hecke operators and will lift through the
Shimura correspondence to Fχ−1 . By a theorem of Waldspurger [26] we will be able to
equate the vanishing of the central critical values of the quadratic twists of L(F, s) to
the vanishing of certain Fourier coefficients of f . Since our ternary forms Q1 and Q2 are
the only forms in a certain genus of ternary forms, we are able to study the automorph
structure of these forms to show that the Fourier coefficients of f are related modulo
3 to certain class numbers of imaginary quadratic number fields. We will then use the
Davenport-Heilbronn theorem (see [18]) to show that at least 7/64 of these class numbers
are not divisible by 3, and hence, the associated Fourier coefficients of f are nonzero. It
will then follow that at least 7/64 of the quadratic twists of L(F, s) have nonzero central
critical value.
The key ingredient in the above argument is that our ternary forms Q1 and Q2 are
a complete set of representatives for a genus of forms having the correct automorph
structure. In particular, if Ai denotes the number of automorphs of Qi (i=1,2), then we
need that A1 + A2 ≡ 0 modulo 3 while 3 ∤ A1 A2 . This congruence modulo 3 is what
allows the use of the Davenport-Heilbronn Theorem. It is important to note that there
are examples of genera of ternary quadratic forms which contain exactly two equivalence
classes but whose automorph structure does not satisfy the above congruence modulo 3
(see the tables of ternary forms in [14]) and, for such examples the Davenport-Heilbronn
Theorem is of no consequence.
2. BACKGROUND
The theory developed by Waldspurger in [26] provides a tool for obtaining information
about the central critical values of the L-series L(fχn , s) associated to the quadratic
twists of a particular integral weight newform f . Before stating his results we need to
introduce one more bit of notation. If f is a newform of weight 2k and if χ is a Dirichlet
character, then fχ is an eigenform for all of the Hecke operators. Hence, by the theory of
newforms developed in [1] and [15], there exists a newform of weight 2k which, following
Waldspurger, we will denote f ·χ with the same eigenvalues as fχ for all but finitely many
of the Hecke operators. In fact it is the central critical values of the L(f · χn , s) which
Waldspurger’s theorem allows us to relate to the Fourier coefficients of a half-integral
weight form. Since fχn and f · χn have the same eigenvalues for all but a finite number
of the Hecke operators, it follows that L(f · χn , s) and L(fχn , s) differ only by a finite
number of Euler factors. Thus, L(fχn , k) = 0 if and only if L(f · χn , k) = 0.
Now, we are ready to state a special case of the main theorem in [26].
4
KEVIN JAMES
Theorem 2.1. Let k ≥ 3 be an odd integer, N ∈ 4N, χ a Dirichlet character modulo
N , and M some divisor of N so that χ2 is a Dirichlet character modulo M . Suppose
F ∈ Sk−1 (M, χ2 ) is a newform with Hecke eigenvalues λp (F ). Suppose also that there
exists a cusp form f ∈ Sk/2 (N, χ) having the property that for all but finitely many
primes p, Tp f = λp (F )f . Finally suppose that the Dirichlet character ν defined by
k−1
2
has conductor divisible by 4. Let Nsf denote the square-free natural
ν(n) = χ(n)( −1
n )
numbers. Then there is a function A : Nsf → C, depending only on F and satisfying the
following condition.
(A(t))2 = L(F · ν −1 χt ,
k−1
) · ǫ(ν −1 χt , 1/2),
2
where ǫ(ψ, s) is chosen so that if L(ψ, s) is the Dirichlet L-function for the Dirichlet
character ψ and if
Λ(ψ, s) =
(
π −s/2 Γ( 2s )L(ψ, s)
if ψ(−1) = 1
π −(s+1)/2 Γ( s+1
2 )L(ψ, s) if ψ(−1) = −1
then
Λ(ψ −1 , 1 − s) = ǫ(ψ, s)Λ(ψ, s).
Moreover f can be written as a finite C-linear combination of Hecke eigenforms fi such
that at (fi ) = c(tsf , F )A(t), where tsf denotes the square-free part of t and c(tsf , F ) ∈ C.
In particular, we can deduce from Theorem 2.1 that if at (f ) =
6
0 then
)
=
6
0.
L(F · ν −1 χt , k−1
2
Also, we will use the following theorem which is an immediate corollary of a theorem
of Davenport and Heilbronn [4] as improved by Nakagawa and Horie [18].
Theorem 2.3. Suppose that m and N satisfy:
1. If p is an odd prime dividing (N, m) then p2 | N and p2 ∤ m, and
2. If N is even, then either 4 | N and m ≡ 1 modulo 4 or 16 | N and m ≡ 8 or 12
modulo 16.
Let T denote the set of discriminants ∆ of imaginary quadratic extensions of Q in the
arithmetic progression ∆ ≡ m modulo N . Then there is a subset S of T having lower
density at least 21 in T such that if ∆ ∈ S then 3 ∤ h(∆).
3. NONVANISHING THEOREMS
If Q is a primitive positive definite ternary quadratic
then we will denote the
P form,Q(x,y,z)
discriminant of Q by dQ . We also define θQ (τ ) = x,y,z∈Z q
(q = e2πiτ ). It is
well known (see [24]) that θQ is a modular form of weight 3/2 and, we have a theorem
of Siegel [25] which states that if Q1 and Q2 are two primitive positive definite ternary
quadratic forms belonging to the same genus then (θQ1 − θQ2 ) is a cusp form.
L-SERIES WITH NON-ZERO CENTRAL CRITICAL VALUE
5
Proposition 3.1. Suppose that Q1 and Q2 are even integral primitive positive definite
ternary quadratic forms and that Q1 and Q2 are the only forms in a genus of forms.
Let Ai denote the number of automorphs of Qi (i = 1, 2). Assume that 3 ∤ A1 A2 but
3 | A1 + A2 . Suppose also that f = (θQ1 − θQ2 ) ∈ S3/2 (N, χq ) is a Hecke-eigenform which
lifts through the Shimura correspondence to a cusp form F ∈ S2 (N/2). Let G denote the
unique weight 2 newform of trivial character having λp (F ) = λp (G) for all but finitely
many of the primes p and, let NG denote the level of G. Put
∗
sf
R = a ∈ (Z/4dsf
and,
Q1 Z) : ∃ a square-free n ≡ a (mod 4dQ1 ) with 3 ∤ an (f )
#R
(4)
δ = sf Q
1 .
8dQ1 p|dsf (1 − p2 )
Q1
Then, the set of square-free natural numbers n such that L(G · χ−qn , 1) 6= 0 has lower
density at least δ in the square-free natural numbers.
Proof. Let R(Q1 , m) denote the number of essentially distinct primitive representations
of m by the genus of ternary forms containing Q1 and let ri (m) denote the total number
of representations of m by Qi (i = 1, 2). Then for sufficiently large square-free natural
numbers m, we have R(Q1 , m) = r1A(m)
+ r2A(m)
.
1
2
Now, suppose that a ∈ R. Then there exists n ≡ a modulo 4dsf
Q1 such that 3 ∤ an (f )
hence, it follows from the construction of f that R(Q1 .n) 6= 0. Thus, R(Q1 , m) 6= 0 for all
natural numbers m ≡ a modulo 4dsf
Q1 . Applying a theorem of Gauss (see [12 Theorem 86])
and recalling the relationship of the class number of an order in an imaginary quadratic
field to the class number of the ring of integers in the same field, we have that for all
square-free natural numbers m ≡ a modulo 4dsf
Q1 ,
R(Q1 , m) = ρh(∆−m ),
(7)
where ρ ∈ Q. Since 3 ∤ A1 A2 and 3 | A1 + A2 , we have that A1 A2 R(Q1 , m) = A2 r1 (m) +
A1 r2 (m) ≡ A2 (r1 (m) − r2 (m)) modulo 3. From our construction of f , we have that
am (f ) = r1 (m) − r2 (m). Therefore, 3 | am (f ) if and only if 3 | ρh(∆−m ). Recall
that 3 ∤ an (f ) and n ≡ a modulo 4dsf
Q1 and therefore, 3 ∤ ρh(∆−n ). Thus, we see that
ord3 (ρ) ≤ 0. By the Davenport-Heilbronn Theorem (Theorem 2.3), we have for at least
half of the square-free natural numbers m ≡ a modulo 4dsf
Q1 , that 3 ∤ h(∆−m ). Therefore,
we have that ord3 (ρ) = 0 and hence it follows for all square-free natural numbers m ≡ a
modulo 4dsf
Q1 , that 3 | am (f ) if and only if 3 | h(∆−m ). Now, applying Theorem 2.3
again, we see for each a ∈ R, that for at least 1/2 of the square-free natural numbers
m ≡ a modulo 4dsf
Q1 , am (G) 6= 0 and hence by Theorem 2.1 that L(G · χ−qm , 1) 6= 0.
sf 2
We note that each a ∈ R gives rise to dsf
Q1 arithmetic progressions modulo 4(dQ1 ) ,
2
and that the total number of arithmetic progressions modulo 4(dsf
Q1 ) in which squareQ
1
2
free numbers reside is 4(dsf
Q1 )
p|dsf (1 − p2 ). Thus the density of square-free natural
Q1
numbers m which are congruent modulo 4dsf
Q1 to some a ∈ R is
proposition now follows from Theorem 2.3.
#R·dsf
Q Q1
2
4(dsf
Q1 )
p|dsf
Q1
(1−
1
p2
)
. The
6
KEVIN JAMES
Example 3.1 Let
Q1 (x, y, z) = x2 + 7y 2 + 7z 2 , and
(8)
Q2 (x, y, z) = 2x2 + 4y 2 + 7z 2 − 2xy.
Then one can easily check that Q1 and Q2 both have discriminant 196 and that they
are both in the same genus. The numbers of automorphs of Q1 and Q2 are 8 and 4
respectively. Also, we can calculate (see [14] or [24]) that θQ1 , θQ2 ∈ M3/2 (28), and
therefore by [25] we have that f = (θQ1 − θQ2 ) ∈ S3/2 (28). We checked computationally
that f is an eigenform for all of the Hecke operators and that f lifts through the Shimura
correspondence to twice the weight 2 new form F of level 14 associated to the elliptic
curve E : y 2 = x3 + x2 + 72x − 368 of conductor 14, that is L(F, s) = L(E, s). Thus,
f satisfies the hypotheses of Proposition 3.1. In this case, we have dsf
Q1 = 14, and by
computing the first 200 coefficients of f , we see that 1, 9, 15, 23, 25, 29, 37, 39 and 53
are all elements of R. Therefore, δ = 7/64 and, Theorem 1 now follows from Proposition
3.1.
We were also able to obtain positive density nonvanishing results as in Corollary 2
for the quadratic twists of nine other elliptic curves. Since the calculations involved in
the verification of the hypotheses of Proposition 3.1 are completely analogous to the
calculations discussed in Example 3.1, we omit them and simply present the results in
the following table. We list for each curve E a Weierstrauss equation for E, the conductor
NE of E, and the lower bound δE on the lower density of square-free natural numbers d
such that L(E−d , 1) 6= 0.
E
y 2 = x3 + x2 + 72x − 368
y 2 = x3 + 8
y 2 = x3 + 1
2
y = x3 + 4x2 − 144x − 944
y 2 = x3 + x2 + 4x + 4
y 2 = x3 + x2 − 72x − 496
y 2 = x3 + x2 + 24x + 144
y 2 = x3 + x2 − 48x + 64
y 2 = x3 + x2 + 3x − 1
y 2 = x3 + 5x2 − 200x − 14000
NE
14
576
36
19
20
26
30
34
44
50
δE
7/64
1/4
5/24
19/240
5/72
13/112
5/128
17/144
11/144
5/24
ACKNOWLEDGMENTS
The author wishes to thank Ken Ono for many insightful discussions and for the
suggestion of attacking this problem by studying the automorph structure of ternary
forms. He would also like to thank Andrew Granville and David Penniston for their help
during the preparation of this manuscript.
L-SERIES WITH NON-ZERO CENTRAL CRITICAL VALUE
7
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